Categorical Extension of Dualities: From Stone to de Vries and Beyond, I

Categorical Extension of Dualities: From Stone to de Vries and Beyond, II

“…and an embedding J of X as a full subcategory of a category Y, the general categorical extension theorem of [11] provides a natural construction for a category B into which A may be fully embedded via I, and which allows for an extension along I and J of the dual equivalence between A and X to a dual equivalence between B and Y. The construction depends on a given X-covering class P of morphisms in Y, defined to satisfy the following conditions:…”

“…Construction 3.2. [11] For any functor T : A op → X and an X-covering class P in Y, we form the (comma) category C(A, P, X), defined as follows:…”

“…We note that, with the help of a simplified variant (obtained in [10]) of our general categorical procedure for the extension of dual equivalences from [11], we gave in [10] a new proof of the Fedorchuk Duality Theorem [16]. Other applications of the general categorical procedure for the extension of dual equivalences will be presented in [12].…”

“…[11] Let X be a full subcategory of Y and P an X-covering class in Y. Then any dual equivalence A op T I and J, with J : X ֒→ Y the inclusion functor and I the full embedding A → B := C(A, P, X)/ ∼, being defined as above.…”

A Unified Approach to Gelfand and de Vries Dualities

“…In [12, 13] both dualities were extended to completely regular spaces and their compactifications. In [21] Gelfand duality was generalized to the setting of compact ordered spaces studied by Nachbin [33], and in [25] a general categorical framework was developed that yields de Vries duality and its generalizations. However, as far as we know, there is no unifying approach to Gelfand and de Vries dualities.…”